Theorem clnbgrssvtx 48480

Description: The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)

Hypothesis

Ref	Expression
clnbgrvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrssvtx	⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem clnbgrssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrisvtx 48479	. 2 ⊢ (𝑛 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3949	1 ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Syntax hints:   = wceq 1567   ⊆ wss 3913  ‘cfv 6534  (class class class)co 7408  Vtxcvtx 29283   ClNeighbVtx cclnbgr 48467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-clnbgr 48468

This theorem is referenced by:  clnbgrlevtx  48494  clnbgrisubgrgrim  48581  clnbgrgrim  48583  isubgr3stgrlem6  48620  isubgr3stgrlem7  48621  isubgr3stgrlem8  48622  isubgr3stgr  48624  uhgrimgrlim  48636  grlicref  48661  grlicsym  48662  clnbgr3stgrgrlim  48668  clnbgr3stgrgrlic  48669